Positivity (2014) 18:219–221
A remark on the paper “Laterally closed lattice
Received: 21 November 2012 / Accepted: 25 April 2013 / Published online: 8 May 2013
© Springer Basel 2013
Abstract A new and simple proof of the main result of the paper “Laterally closed
lattice homomorphisms” by Toumi and Toumi (J Math Anal Appl 324:1178–1194,
2006) is given following the paper “Extension of Riesz homomorphisms, I” by Buskes
(J Aust Math Soc Ser A 39(1):107–120, 1985).
Keywords Riesz space · Laterally closed homomorphism · Universal completion
Mathematics subject Classiﬁcation (2000) Primary 46A40
All Riesz spaces in this paper are assumed to be Archimedean. We do not repeat the
deﬁnition and notations of the well-known terminology. We will follow the standard
terminology of [1,2] and . For a Riesz space E, the Dedekind completion of E is
denoted by E
, and the universal completion of E is denoted by E
.IfD ⊂ E, then
[D] denotes the band of E generated by D.
The following deﬁnitions are given .
Deﬁnition 1.1 (i) Let I, E and F be Riesz spaces with I an ideal of E.ARiesz
homomorphism π : I → F is called o(I, E, F) -continuous if for each x ∈ E,
the set π([0, x]∩I ) is order bounded in E.
(ii) Let E and F be Riesz spaces. The pair (E , F ) is said to have the (oI)-property if
every o(I, E, F)-continuous Riesz homomorphism π : I → E has an extension
π : E → F, which is a Riesz homomorphism.
Z. Ercan (
Department of Mathematics, Abant
Izzet Baysal University, Gölköy Kampüsü, Bolu, Turkey